### Abstract

The free entropies χ̂(a_{1},...,a_{N}) of non-selfadjoint random variables and χ_{u}(u_{1},...,u_{N}) of unitary random variables are introduced and discussed by the methods of Voiculescu's free analysis. The additivity χ_{u}(u_{1},...,u_{N}) = ∑_{i} χ_{u}(u_{i}) is shown to be equivalent to freeness. The relation among χ̂, χ_{u} and χ is investigate in the case when a_{i} = u_{i}h_{i} is the polar decomposition. The subadditivity χ̂(a^{1},...,a_{N}) <χ_{u}(u_{1},...,u_{N}) + χ(h^{2}_{1},...,h^{2}_{N}) + constant is proven and applications to some maximization problems for χ̂ are given.

Original language | English |
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Pages (from-to) | 421-444 |

Number of pages | 24 |

Journal | Communications in Mathematical Physics |

Volume | 202 |

Issue number | 2 |

Publication status | Published - Apr 1999 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Communications in Mathematical Physics*,

*202*(2), 421-444.

**Properties of free entropy related to polar decomposition.** / Hiai, Fumio; Petz, D.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 202, no. 2, pp. 421-444.

}

TY - JOUR

T1 - Properties of free entropy related to polar decomposition

AU - Hiai, Fumio

AU - Petz, D.

PY - 1999/4

Y1 - 1999/4

N2 - The free entropies χ̂(a1,...,aN) of non-selfadjoint random variables and χu(u1,...,uN) of unitary random variables are introduced and discussed by the methods of Voiculescu's free analysis. The additivity χu(u1,...,uN) = ∑i χu(ui) is shown to be equivalent to freeness. The relation among χ̂, χu and χ is investigate in the case when ai = uihi is the polar decomposition. The subadditivity χ̂(a1,...,aN) <χu(u1,...,uN) + χ(h21,...,h2N) + constant is proven and applications to some maximization problems for χ̂ are given.

AB - The free entropies χ̂(a1,...,aN) of non-selfadjoint random variables and χu(u1,...,uN) of unitary random variables are introduced and discussed by the methods of Voiculescu's free analysis. The additivity χu(u1,...,uN) = ∑i χu(ui) is shown to be equivalent to freeness. The relation among χ̂, χu and χ is investigate in the case when ai = uihi is the polar decomposition. The subadditivity χ̂(a1,...,aN) <χu(u1,...,uN) + χ(h21,...,h2N) + constant is proven and applications to some maximization problems for χ̂ are given.

UR - http://www.scopus.com/inward/record.url?scp=0033238571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033238571&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033238571

VL - 202

SP - 421

EP - 444

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -